This document discusses different measures of position and distribution in statistics including standard scores (z-scores), quartiles, deciles, and percentiles. It defines standard scores as measuring how many standard deviations a value is from the mean. Quartiles divide a sorted data set into four equal parts, with the lower and upper quartiles marking the boundaries between the lower/upper half of data. Deciles divide data into 10 equal parts. Percentiles indicate the percentage of values in a data set that fall below a given value. The document provides examples of how to calculate and interpret these different measures using sample data sets.

1. Measure of Position Looking at individual values: Where does it lie among the sample (or population) standard score (or z score) quartiles, deciles, and percentiles 25%-75% SAT scores Princeton 1370 – 1560 Villanova 1180 – 1340

2. Standard Score (z score) How many standard deviations from the mean is a sample? Z scores less that –2.00 or greater that 2.00 are outliers.

3. For example 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100 Mean : 77 Standard deviation = 18 mean mean +  mean + 2  mean + 3  mean - 3  mean - 2  mean - 1 

4. Quartiles Separates the sorted list into four parts separated by three boundaries min lower quartile median max upper quartile 25% 25% 25% 25% If the median separates the list into two halves The lower quartile is the median of the lower half The upper quartile is the median of the upper half

5. For example 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

6. Deciles 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100 min median max 10% 10% 10%

7. Percentile Percentile value of a value x Where does x sit in a set of data For example: 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100 What is the percentile of:

8. Finding the value at the k th Percentile n values in our sample Data has to be sorted Compute L, the position of the k th percentile If L is a whole number, the percentile is midway between L and L+1 If L is a decimal, round up; the percentile is the L th value

9. For example 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100 What is the value at the

10. Relationships The 1 st quartile is the same as the 25 th percentile The 3 rd quartile is the same as the 75 th percentile The first decile is the same as the 10 th percentile

11. Live example What are the mean and standard deviation What are the minimum, lower quartile, median, upper quartile, and maximum What is the 4 th decile What is the 35% percentile? What is South’s z value What is South’s percentile?

12. Your turn What are the mean and standard deviation What are the minimum, lower quartile, median, upper quartile, and maximum What is the 4 th decile? What is the 35% percentile? What is Rob’s z score? What is Rob’s percentile?